3. 1st Order Spatial Point Patterns Analysis Methods

Author

Santhya Selvan

Published

August 30, 2024

Modified

September 2, 2024

3.1 Overview

Spatial Point Pattern Analysis helps us to evaluate the patter or distribution of a set of points. The points can be locations of events (like accidents, crimes, etc.) and places of business (like retail, services etc.)

In this hands-on, I will look into the spatial point processes of childcare centers in Singapore with the help of suitable functions from of spatstat. Mainly, this exercise seeks to find:

  • If the childcare centres in Singapore randomly distributed throughout the country,

  • If no, then where are the locations with higher concentration of childcare centres?

Let’s start exploring :)

3.2 The data

In this exercise, I will be making use of these datasets:

  • CHILDCARE, a point feature data providing both location and attribute information of childcare centres. It was downloaded from Data.gov.sg and is in geojson format.

  • MP14_SUBZONE_WEB_PL, a polygon feature data providing information of URA 2014 Master Plan Planning Subzone boundary data. It is in ESRI shapefile format. This data set was also downloaded from Data.gov.sg.

  • CostalOutline, a polygon feature data showing the national boundary of Singapore. It is provided by SLA and is in ESRI shapefile format.

3.3 Installing and Loading the R packages

In this hands-on exercise, five R packages will be used:

  • sf, a relatively new R package specially designed to import, manage and process vector-based geospatial data in R.

  • spatstat, which has a wide range of useful functions for point pattern analysis. In this hands-on exercise, it will be used to perform 1st- and 2nd-order spatial point patterns analysis and derive kernel density estimation (KDE) layer.

  • raster which reads, writes, manipulates, analyses and model of gridded spatial data (i.e. raster). In this hands-on exercise, it will be used to convert image output generate by spatstat into raster format.

  • maptools which provides a set of tools for manipulating geographic data. In this hands-on exercise, we mainly use it to convert Spatial objects into ppp format of spatstat.

  • tmap which provides functions for plotting cartographic quality static point patterns maps or interactive maps by using leaflet API.

Let’s install them first

pacman::p_load(sf, spatstat, raster, tidyverse, tmap)

3.4 Data Wrangling

3.4.1 Importing the Spatial Data

I will use st_read() of sf package to import the three geospatial data sets into R.

childcare_sf <- st_read("data4/child-care-services-geojson.geojson") %>% st_transform(crs = 3414)
Reading layer `child-care-services-geojson' from data source 
  `C:\santhyats\IS415-GAA\Hands-on_Ex\Hands-on_Ex03\data4\child-care-services-geojson.geojson' 
  using driver `GeoJSON'
Simple feature collection with 1545 features and 2 fields
Geometry type: POINT
Dimension:     XYZ
Bounding box:  xmin: 103.6824 ymin: 1.248403 xmax: 103.9897 ymax: 1.462134
z_range:       zmin: 0 zmax: 0
Geodetic CRS:  WGS 84
coastal_sf <- st_read(dsn = "data4/",
                      layer = "CostalOutline")
Reading layer `CostalOutline' from data source 
  `C:\santhyats\IS415-GAA\Hands-on_Ex\Hands-on_Ex03\data4' using driver `ESRI Shapefile'
Simple feature collection with 60 features and 4 fields
Geometry type: POLYGON
Dimension:     XY
Bounding box:  xmin: 2663.926 ymin: 16357.98 xmax: 56047.79 ymax: 50244.03
Projected CRS: SVY21
mpsz_sf <- st_read(dsn = "data4/",
                   layer = "MP14_SUBZONE_WEB_PL")
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `C:\santhyats\IS415-GAA\Hands-on_Ex\Hands-on_Ex03\data4' using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

Before doing further analysis, I will check the information of the reference system of the data sets so that we can make changes (if necessary) and ensure that all the datasets make use of the proper crs.

st_crs(childcare_sf)
Coordinate Reference System:
  User input: EPSG:3414 
  wkt:
PROJCRS["SVY21 / Singapore TM",
    BASEGEOGCRS["SVY21",
        DATUM["SVY21",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["degree",0.0174532925199433]],
        ID["EPSG",4757]],
    CONVERSION["Singapore Transverse Mercator",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["northing (N)",north,
            ORDER[1],
            LENGTHUNIT["metre",1]],
        AXIS["easting (E)",east,
            ORDER[2],
            LENGTHUNIT["metre",1]],
    USAGE[
        SCOPE["Cadastre, engineering survey, topographic mapping."],
        AREA["Singapore - onshore and offshore."],
        BBOX[1.13,103.59,1.47,104.07]],
    ID["EPSG",3414]]
st_crs(coastal_sf)
Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]
st_crs(mpsz_sf)
Coordinate Reference System:
  User input: SVY21 
  wkt:
PROJCRS["SVY21",
    BASEGEOGCRS["SVY21[WGS84]",
        DATUM["World Geodetic System 1984",
            ELLIPSOID["WGS 84",6378137,298.257223563,
                LENGTHUNIT["metre",1]],
            ID["EPSG",6326]],
        PRIMEM["Greenwich",0,
            ANGLEUNIT["Degree",0.0174532925199433]]],
    CONVERSION["unnamed",
        METHOD["Transverse Mercator",
            ID["EPSG",9807]],
        PARAMETER["Latitude of natural origin",1.36666666666667,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8801]],
        PARAMETER["Longitude of natural origin",103.833333333333,
            ANGLEUNIT["Degree",0.0174532925199433],
            ID["EPSG",8802]],
        PARAMETER["Scale factor at natural origin",1,
            SCALEUNIT["unity",1],
            ID["EPSG",8805]],
        PARAMETER["False easting",28001.642,
            LENGTHUNIT["metre",1],
            ID["EPSG",8806]],
        PARAMETER["False northing",38744.572,
            LENGTHUNIT["metre",1],
            ID["EPSG",8807]]],
    CS[Cartesian,2],
        AXIS["(E)",east,
            ORDER[1],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]],
        AXIS["(N)",north,
            ORDER[2],
            LENGTHUNIT["metre",1,
                ID["EPSG",9001]]]]

From the results, we notice that only the childcare_sf dataframe has the standard coordinate system. As such, I will be modifying the other two data frames so that all three of them will be consistent.

coastal_sf <- st_transform(coastal_sf, crs = 3414)
mpsz_sf <- st_transform(mpsz_sf, crs = 3414)

3.4.2 Mapping the Geospatial Datasets

We will map the datasets together so we can better see the spatial pattern of the childcare centers.

tmap_mode("plot")
tmap mode set to plotting
tm_shape(coastal_sf) + tm_polygons() + tm_shape(mpsz_sf) + tm_polygons() +
tm_shape(childcare_sf) + tm_dots() 

We are able to plot all the data sets on the same map because we have ensured that their referencing systems are consistent with one another. This allows me to further appreciate the value of ensuring the referencing systems of the datasets before we begin our analysis.

Alternatively, we can also prepare a pin map by using the code chunk below.

tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(childcare_sf)+
  tm_dots()
tmap_mode("plot")
tmap mode set to plotting

3.5 Geospatial Data Wrangling

3.5.1 Converting simple dataframe to spatstat’s ppp object with the as.ppp() function.

We can create the ppp object by providing the point coordinates and the observation window.

childcare_ppp <- as.ppp(st_coordinates(childcare_sf), st_bbox(childcare_sf))
Warning: data contain duplicated points
plot(childcare_ppp)

We can take a look at the summary statistics of the newly created ppp object:

summary(childcare_ppp)
Marked planar point pattern:  1545 points
Average intensity 1.91145e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 11 decimal places

marks are numeric, of type 'double'
Summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      0       0       0       0       0       0 

Window: rectangle = [11203.01, 45404.24] x [25667.6, 49300.88] units
                    (34200 x 23630 units)
Window area = 808287000 square units

We see that there is a warning message indicating that there are duplicates in the pattern. This is a significant issue in spatial point pattern analysis. How do we then handle them?

3.5.2 Handling Duplicates

We can check for duplication using this code:

any(duplicated(childcare_ppp))
[1] TRUE

Since the check has returned TRUE, we will further seek to find the the number of co-indicence points using the  multiplicity() function.

multiplicity(childcare_ppp)
   1    2    3    4    5    6    7    8    9   10   11   12   13   14   15   16 
   1    1    1    3    1    1    1    1    2    1    1    1    1    1    1    1 
  17   18   19   20   21   22   23   24   25   26   27   28   29   30   31   32 
   1    1    1    1    1    1    1    1    1    1    9    1    1    1    1    1 
  33   34   35   36   37   38   39   40   41   42   43   44   45   46   47   48 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
  49   50   51   52   53   54   55   56   57   58   59   60   61   62   63   64 
   1    1    1    1    1    1    2    1    1    3    1    1    1    1    1    1 
  65   66   67   68   69   70   71   72   73   74   75   76   77   78   79   80 
   1    1    1    1    1    2    1    1    1    1    1    2    1    1    1    1 
  81   82   83   84   85   86   87   88   89   90   91   92   93   94   95   96 
   1    1    1    3    1    1    1    1    1    1    1    1    1    1    1    1 
  97   98   99  100  101  102  103  104  105  106  107  108  109  110  111  112 
   1    1    1    1    1    1    1    1    2    1    1    1    1    1    1    1 
 113  114  115  116  117  118  119  120  121  122  123  124  125  126  127  128 
   1    1    1    1    1    1    2    1    1    1    3    1    1    1    2    1 
 129  130  131  132  133  134  135  136  137  138  139  140  141  142  143  144 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    3    2 
 145  146  147  148  149  150  151  152  153  154  155  156  157  158  159  160 
   1    2    1    1    1    2    2    3    1    5    1    5    1    1    1    2 
 161  162  163  164  165  166  167  168  169  170  171  172  173  174  175  176 
   1    1    1    1    2    1    1    1    1    1    1    2    1    1    1    1 
 177  178  179  180  181  182  183  184  185  186  187  188  189  190  191  192 
   1    4    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 193  194  195  196  197  198  199  200  201  202  203  204  205  206  207  208 
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 209  210  211  212  213  214  215  216  217  218  219  220  221  222  223  224 
   2    1    1    1    1    1    1    1    1    1    1    1    2    1    1    1 
 225  226  227  228  229  230  231  232  233  234  235  236  237  238  239  240 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 241  242  243  244  245  246  247  248  249  250  251  252  253  254  255  256 
   1    1    1    1    2    1    1    1    1    1    1    1    1    1    1    1 
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   1    1    1    1    1    1    1    9    1    1    2    1    1    1    1    1 
 305  306  307  308  309  310  311  312  313  314  315  316  317  318  319  320 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
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 353  354  355  356  357  358  359  360  361  362  363  364  365  366  367  368 
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 417  418  419  420  421  422  423  424  425  426  427  428  429  430  431  432 
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   1    1    1    1    2    1    1    1    1    1    1    1    1    1    1    1 
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   1    1    9    9    1    1    1    1    1    1    1    1    1    1    2    1 
 465  466  467  468  469  470  471  472  473  474  475  476  477  478  479  480 
   2    1    1    1    1    1    1    1    1    1    1    1    2    2    1    1 
 481  482  483  484  485  486  487  488  489  490  491  492  493  494  495  496 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 497  498  499  500  501  502  503  504  505  506  507  508  509  510  511  512 
   1    1    1    1    1    1    2    1    1    1    1    1    1    1    1    2 
 513  514  515  516  517  518  519  520  521  522  523  524  525  526  527  528 
   1    1    1    1    1    1    1    1    1    1    1    2    1    1    3    1 
 529  530  531  532  533  534  535  536  537  538  539  540  541  542  543  544 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 545  546  547  548  549  550  551  552  553  554  555  556  557  558  559  560 
   1    1    1    1    1    1    1    1    1    3    1    1    1    1    1    1 
 561  562  563  564  565  566  567  568  569  570  571  572  573  574  575  576 
   2    2    2    1    1    1    1    2    1    1    2    1    1    1    2    1 
 577  578  579  580  581  582  583  584  585  586  587  588  589  590  591  592 
   1    2    1    1    1    1    1    9    1    4    1    2    1    1    1    1 
 593  594  595  596  597  598  599  600  601  602  603  604  605  606  607  608 
   2    1    1    1    1    1    1    1    2    1    2    1    1    1    1    1 
 609  610  611  612  613  614  615  616  617  618  619  620  621  622  623  624 
   1    1    1    1    1    1    1    1    1    2    1    2    1    1    1    1 
 625  626  627  628  629  630  631  632  633  634  635  636  637  638  639  640 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
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 657  658  659  660  661  662  663  664  665  666  667  668  669  670  671  672 
   1    1    1    1    1    1    1    3    1    1    1    1    1    1    1    1 
 673  674  675  676  677  678  679  680  681  682  683  684  685  686  687  688 
   1    1    1    1    1    4    1    1    1    1    1    4    1    1    1    1 
 689  690  691  692  693  694  695  696  697  698  699  700  701  702  703  704 
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 705  706  707  708  709  710  711  712  713  714  715  716  717  718  719  720 
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 721  722  723  724  725  726  727  728  729  730  731  732  733  734  735  736 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
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   1    2    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
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 769  770  771  772  773  774  775  776  777  778  779  780  781  782  783  784 
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 785  786  787  788  789  790  791  792  793  794  795  796  797  798  799  800 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 801  802  803  804  805  806  807  808  809  810  811  812  813  814  815  816 
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 817  818  819  820  821  822  823  824  825  826  827  828  829  830  831  832 
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 833  834  835  836  837  838  839  840  841  842  843  844  845  846  847  848 
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 849  850  851  852  853  854  855  856  857  858  859  860  861  862  863  864 
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   3    1    1    1    2    1    1    1    3    1    1    3    1    1    1    1 
 897  898  899  900  901  902  903  904  905  906  907  908  909  910  911  912 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 913  914  915  916  917  918  919  920  921  922  923  924  925  926  927  928 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 929  930  931  932  933  934  935  936  937  938  939  940  941  942  943  944 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 945  946  947  948  949  950  951  952  953  954  955  956  957  958  959  960 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
 961  962  963  964  965  966  967  968  969  970  971  972  973  974  975  976 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 977  978  979  980  981  982  983  984  985  986  987  988  989  990  991  992 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
 993  994  995  996  997  998  999 1000 1001 1002 1003 1004 1005 1006 1007 1008 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 
   1    1    1    1    1    1    1    1    1    2    2    1    1    1    1    1 
1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 
   1    1    1    1    1    2    1    1    1    1    1    1    1    1    1    1 
1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 
   1    1    1    1    1    1    1    1    2    2    1    1    1    5    1    1 
1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 
   1    1    1    1    1    1    1    1    1    2    1    1    1    1    1    1 
1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    1 
1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 
   1    9    1    2    2    1    1    1    2    1    1    1    1    1    1    1 
1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 
   1    1    1    1    2    1    1    1    3    1    1    1    1    1    1    1 
1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 
   9    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 
   1    1    1    2    1    1    1    1    1    1    1    1    1    1    1    1 
1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    2 
1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 
   1    1    1    2    1    2    1    1    1    2    2    2    1    1    1    1 
1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 
   1    1    2    1    1    1    1    1    1    1    1    1    2    1    1    1 
1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 
   1    1    1    1    3    1    1    1    1    1    1    1    1    1    1    1 
1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 
   1    1    1    1    1    1    1    1    4    1    1    1    1    1    2    1 
1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 
   1    1    1    1    1    1    1    1    1    9    1    1    1    1    1    1 
1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    2    1 
1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 
   1    2    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 
   1    1    1    1    1    1    1    1    1    1    2    1    1    1    1    1 
1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 
   1    1    1    1    1    1    2    1    1    1    1    1    1    1    1    1 
1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 
   1    1    1    1    1    1    1    1    1    1    5    1    1    1    1    1 
1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 
   1    1    1    1    1    1    1    1    1    1    1    1    1    1    1    1 
1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 
   1    1    1    1    1    2    1    1    1    1    2    1    1    1    1    3 
1537 1538 1539 1540 1541 1542 1543 1544 1545 
   1    1    1    1    1    1    2    1    1 

We can also find out how man locations have more than one point events.

sum(multiplicity(childcare_ppp) > 1)
[1] 128

From the output, we see that there are 128 duplicated point events. To see the locations of these, we will plot them out.

tmap_mode('view')
tmap mode set to interactive viewing
tm_shape(childcare_sf) +
  tm_dots(alpha=0.4, 
          size=0.05)
tmap_mode("plot")
tmap mode set to plotting

There are three ways to overcome this problem:

  • We can easily delete the duplicates. But, that will also mean that some useful point events will be lost.

  • The second solution is use jittering, which will add a small perturbation to the duplicate points so that they do not occupy the exact same space.

  • The third solution is to make each point “unique” and then attach the duplicates of the points to the patterns as marks, as attributes of the points. Then you would need analytical techniques that take into account these marks.

The code chunk below implements the jittering approach.

childcare_ppp_jit <- rjitter(childcare_ppp, 
                             retry=TRUE, 
                             nsim=1, 
                             drop=TRUE)

We can once again check if there are any duplicates in our newly created dataset.

any(duplicated(childcare_ppp_jit))
[1] FALSE

3.5.3 Creating owin object

When analysing spatial point patterns, it is a good practice to confine the analysis with a geographical area like Singapore boundary. In spatstat, an object called owin is specially designed to represent this polygonal region.

Because we already have the dataframe for Singapore’s Coastline, we will use that (coastal_sf) and convert it into an owin object.

coastal_owin <- as.owin(coastal_sf)
plot(coastal_owin)

Here is the summary for the owin object

summary(coastal_owin)
Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

3.5.6 Combining point events object and owin object

In this last step of geospatial data wrangling, we will extract childcare events that are located within Singapore.

childcareSG_ppp = childcare_ppp[coastal_owin]
summary(childcareSG_ppp)
Marked planar point pattern:  1545 points
Average intensity 2.129929e-06 points per square unit

*Pattern contains duplicated points*

Coordinates are given to 11 decimal places

marks are numeric, of type 'double'
Summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
      0       0       0       0       0       0 

Window: polygonal boundary
50 separate polygons (1 hole)
                 vertices         area relative.area
polygon 1 (hole)       30     -7081.18     -9.76e-06
polygon 2              55     82537.90      1.14e-04
polygon 3              90    415092.00      5.72e-04
polygon 4              49     16698.60      2.30e-05
polygon 5              38     24249.20      3.34e-05
polygon 6             976  23344700.00      3.22e-02
polygon 7             721   1927950.00      2.66e-03
polygon 8            1992   9992170.00      1.38e-02
polygon 9             330   1118960.00      1.54e-03
polygon 10            175    925904.00      1.28e-03
polygon 11            115    928394.00      1.28e-03
polygon 12             24      6352.39      8.76e-06
polygon 13            190    202489.00      2.79e-04
polygon 14             37     10170.50      1.40e-05
polygon 15             25     16622.70      2.29e-05
polygon 16             10      2145.07      2.96e-06
polygon 17             66     16184.10      2.23e-05
polygon 18           5195 636837000.00      8.78e-01
polygon 19             76    312332.00      4.31e-04
polygon 20            627  31891300.00      4.40e-02
polygon 21             20     32842.00      4.53e-05
polygon 22             42     55831.70      7.70e-05
polygon 23             67   1313540.00      1.81e-03
polygon 24            734   4690930.00      6.47e-03
polygon 25             16      3194.60      4.40e-06
polygon 26             15      4872.96      6.72e-06
polygon 27             15      4464.20      6.15e-06
polygon 28             14      5466.74      7.54e-06
polygon 29             37      5261.94      7.25e-06
polygon 30            111    662927.00      9.14e-04
polygon 31             69     56313.40      7.76e-05
polygon 32            143    145139.00      2.00e-04
polygon 33            397   2488210.00      3.43e-03
polygon 34             90    115991.00      1.60e-04
polygon 35             98     62682.90      8.64e-05
polygon 36            165    338736.00      4.67e-04
polygon 37            130     94046.50      1.30e-04
polygon 38             93    430642.00      5.94e-04
polygon 39             16      2010.46      2.77e-06
polygon 40            415   3253840.00      4.49e-03
polygon 41             30     10838.20      1.49e-05
polygon 42             53     34400.30      4.74e-05
polygon 43             26      8347.58      1.15e-05
polygon 44             74     58223.40      8.03e-05
polygon 45            327   2169210.00      2.99e-03
polygon 46            177    467446.00      6.44e-04
polygon 47             46    699702.00      9.65e-04
polygon 48              6     16841.00      2.32e-05
polygon 49             13     70087.30      9.66e-05
polygon 50              4      9459.63      1.30e-05
enclosing rectangle: [2663.93, 56047.79] x [16357.98, 50244.03] units
                     (53380 x 33890 units)
Window area = 725376000 square units
Fraction of frame area: 0.401

We will plot this newly created object

plot(childcareSG_ppp)

3.6 First-order Spatial Point Patterns Analysis

I will now perform first-order SPPA by using spatstat package. The hands-on exercise will focus on:

  • deriving kernel density estimation (KDE) layer for visualising and exploring the intensity of point processes,

  • performing Confirmatory Spatial Point Patterns Analysis by using Nearest Neighbour statistics.

3.6.1 Kernel Density Estimation

3.6.1.1 Computing kernel density estimation using automatic bandwidth selection method

We can compute a kernel density by using the following configurations of density() of spatstat:

  • bw.diggle() automatic bandwidth selection method. Other recommended methods are bw.CvL(), bw.scott() or bw.ppl().

  • The smoothing kernel used is gaussian, which is the default. Other smoothing methods are: “epanechnikov”, “quartic” or “disc”.

  • The intensity estimate is corrected for edge effect bias by using method described by Jones (1993) and Diggle (2010, equation 18.9). The default is FALSE.

kde_childcareSG_bw <- density(childcareSG_ppp,
                              sigma=bw.diggle,
                              edge=TRUE,
                            kernel="gaussian") 

We will now plot this

plot(kde_childcareSG_bw)

The density values of the output range from 0 to 0.000035 which is way too small to comprehend. This is because the default unit of measurement of svy21 is in meter. As a result, the density values computed is in “number of points per square meter”. In the next section, we will look at how we can change the scale.

3.6.1.2 Rescaling KDE values

We will use the rescale.ppp() function to covert the unit of measurement from meter to kilometer.

childcareSG_ppp.km <- rescale.ppp(childcareSG_ppp, 1000, "km")

Now, we can re-run density() using the resale data set and plot the output kde map.

kde_childcareSG.bw <- density(childcareSG_ppp.km, sigma=bw.diggle, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG.bw)

3.6.2 Working with different automatic bandwidth methods

Beside bw.diggle(), there are three other spatstat functions can be used to determine the bandwidth, they are: bw.CvL(), bw.scott(), and bw.ppl().

Let us take a look at the bandwidth return by these automatic bandwidth calculation methods.

 bw.CvL(childcareSG_ppp.km)
   sigma 
4.543278 
bw.scott(childcareSG_ppp.km)
 sigma.x  sigma.y 
2.224898 1.450966 
bw.ppl(childcareSG_ppp.km)
    sigma 
0.3897114 
bw.diggle(childcareSG_ppp.km)
    sigma 
0.2984095 

Baddeley et. (2016) suggested the use of the bw.ppl() algorithm because in ther experience it tends to produce the more appropriate values when the pattern consists predominantly of tight clusters. But they also insist that if the purpose of once study is to detect a single tight cluster in the midst of random noise then the bw.diggle() method seems to work best.

The code chunk beow will be used to compare the output of using bw.diggle and bw.ppl methods.

kde_childcareSG.ppl <- density(childcareSG_ppp.km, 
                               sigma=bw.ppl, 
                               edge=TRUE,
                               kernel="gaussian")
par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "bw.diggle")
plot(kde_childcareSG.ppl, main = "bw.ppl")

3.6.3 Working with different Kernel Methods

By default, the kernel method used in density.ppp() is gaussian. But there are three other options, namely: Epanechnikov, Quartic and Dics. We can compute three more kernel density estimations using these methods:

par(mfrow=c(2,2))
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="gaussian"), 
     main="Gaussian")
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="epanechnikov"), 
     main="Epanechnikov")
Warning in density.ppp(childcareSG_ppp.km, sigma = bw.ppl, edge = TRUE, :
Bandwidth selection will be based on Gaussian kernel
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="quartic"), 
     main="Quartic")
Warning in density.ppp(childcareSG_ppp.km, sigma = bw.ppl, edge = TRUE, :
Bandwidth selection will be based on Gaussian kernel
plot(density(childcareSG_ppp.km, 
             sigma=bw.ppl, 
             edge=TRUE, 
             kernel="disc"), 
     main="Disc")
Warning in density.ppp(childcareSG_ppp.km, sigma = bw.ppl, edge = TRUE, :
Bandwidth selection will be based on Gaussian kernel

3.7 Fixed and Adaptive KDE

3.7.1 Computing KDE using Fixed bandwidth

Next, we will compute a KDE layer by defining a bandwidth of 600 meter. Notice that in the code chunk below, the sigma value used is 0.6. This is because the unit of measurement of childcareSG_ppp.km object is in kilometer, hence the 600m is 0.6km.

kde_childcareSG_600 <- density(childcareSG_ppp.km, sigma=0.6, edge=TRUE, kernel="gaussian")
plot(kde_childcareSG_600)

3.7.2 Computing KDE using Adaptive bandwidth

Fixed bandwidth method is very sensitive to highly skewed distribution of spatial point patterns. One way to overcome this problem is by using adaptive bandwidth instead.

In this section, you will derive the adaptive kernel density estimation by using density.adaptive() of spatstat.

kde_childcareSG_adaptive <- adaptive.density(childcareSG_ppp.km, method="kernel")  
plot(kde_childcareSG_adaptive)

We can now compare the two maps side-by-side

par(mfrow=c(1,2))
plot(kde_childcareSG.bw, main = "Fixed bandwidth")
plot(kde_childcareSG_adaptive, main = "Adaptive bandwidth")

3.7.3 Converting KDE output into grid object

Although the output will have no difference, we will convert it to a grid object so that it will be more suitable with mapping applications.

gridded_kde_childcareSG_bw <- as(kde_childcareSG.bw, "SpatialGridDataFrame")
spplot(gridded_kde_childcareSG_bw)

3.7.2.1 Converting gridded output to Raster

Next, we will convert the gridded kernal density objects into RasterLayer object by using raster() of raster package.

kde_childcareSG_bw_raster <- raster(kde_childcareSG.bw)

Here are the properties of the raster object:

kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : NA 
source     : memory
names      : layer 
values     : -8.476185e-15, 28.51831  (min, max)

Note that the crs is ‘NA’. Hence, I will assign a projection system to the object.

projection(kde_childcareSG_bw_raster) <- CRS("+init=EPSG:3414")
kde_childcareSG_bw_raster
class      : RasterLayer 
dimensions : 128, 128, 16384  (nrow, ncol, ncell)
resolution : 0.4170614, 0.2647348  (x, y)
extent     : 2.663926, 56.04779, 16.35798, 50.24403  (xmin, xmax, ymin, ymax)
crs        : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +units=m +no_defs 
source     : memory
names      : layer 
values     : -8.476185e-15, 28.51831  (min, max)

3.7.3 Visualising the output in tmap

I will use the code chunk below to visualise the raster object

tm_shape(kde_childcareSG_bw_raster) + 
  tm_raster("layer", palette = "viridis") +
  tm_layout(legend.position = c("right", "bottom"), frame = FALSE)

3.7.4 Comparing Spatial Point Patterns using KDE

In this section, I will compare KDE of childcare at Ponggol, Tampines, Chua Chu Kang and Jurong West planning areas.

We will firstly extract the study areas using this code:

pg <- mpsz_sf %>%
  filter(PLN_AREA_N == "PUNGGOL")
tm <- mpsz_sf %>%
  filter(PLN_AREA_N == "TAMPINES")
ck <- mpsz_sf %>%
  filter(PLN_AREA_N == "CHOA CHU KANG")
jw <- mpsz_sf %>%
  filter(PLN_AREA_N == "JURONG WEST")

We can plot these study areas

par(mfrow=c(2,2))
plot(st_geometry(pg), main = "Ponggol")
plot(st_geometry(tm), main="Tampines")
plot(st_geometry(ck), main="Choa Chu Kang")
plot(st_geometry(jw), main="Jurong West")

3.7.4.1 Creating owin object

Now, we will convert these sf objects into owin objects that is required by spatstat.

pg_owin = as.owin(pg)
tm_owin = as.owin(tm)
ck_owin = as.owin(ck)
jw_owin = as.owin(jw)
3.7.4.2 Combining childcare points and the study area

By using the code chunk below, we are able to extract childcare that is within the specific region to do our analysis later on.

childcare_pg_ppp = childcare_ppp_jit[pg_owin]
childcare_tm_ppp = childcare_ppp_jit[tm_owin]
childcare_ck_ppp = childcare_ppp_jit[ck_owin]
childcare_jw_ppp = childcare_ppp_jit[jw_owin]

Next, rescale.ppp() function is used to trasnform the unit of measurement from metre to kilometre.

childcare_pg_ppp.km = rescale.ppp(childcare_pg_ppp, 1000, "km")
childcare_tm_ppp.km = rescale.ppp(childcare_tm_ppp, 1000, "km")
childcare_ck_ppp.km = rescale.ppp(childcare_ck_ppp, 1000, "km")
childcare_jw_ppp.km = rescale.ppp(childcare_jw_ppp, 1000, "km")

We will now plot these four study areas and their childcare center locations.

par(mfrow=c(2,2))
plot(childcare_pg_ppp.km, main="Punggol")
plot(childcare_tm_ppp.km, main="Tampines")
plot(childcare_ck_ppp.km, main="Choa Chu Kang")
plot(childcare_jw_ppp.km, main="Jurong West")

3.7.4.3 Computing KDE

We will use bw.diggle() to come up with the bandwidth of each.

par(mfrow=c(2,2))
plot(density(childcare_pg_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tempines")
plot(density(childcare_ck_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Choa Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=bw.diggle, 
             edge=TRUE, 
             kernel="gaussian"),
     main="JUrong West")

3.7.4.4 Computing KDE with fixed bandwidth

We will use 250m as the bandwidth.

par(mfrow=c(2,2))
plot(density(childcare_ck_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Chou Chu Kang")
plot(density(childcare_jw_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="JUrong West")
plot(density(childcare_pg_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Punggol")
plot(density(childcare_tm_ppp.km, 
             sigma=0.25, 
             edge=TRUE, 
             kernel="gaussian"),
     main="Tampines")

3.8 Nearest Neighbour Analysis

Here, I will be performing the Clark-Evans test of aggregation for a spatial point pattern by using the clarkevans.test() function of statspat.

The test hypotheses are:

Ho = The distribution of childcare services are randomly distributed.

H1= The distribution of childcare services are not randomly distributed.

The 95% confident interval will be used.

clarkevans.test(childcareSG_ppp,
                correction="none",
                clipregion="sg_owin",
                alternative=c("clustered"),
                nsim=99)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcareSG_ppp
R = 0.55631, p-value < 2.2e-16
alternative hypothesis: clustered (R < 1)

From the results, we are able to see that the spatial points are not randomly distributed and in fact follow a clustered patters, as seen from the extremely small p-value and the R-value being less than 1.

3.8.1 Clark Evans Test for Planning Area: Choa Chu Kang, Tampines

In the code chunk below, clarkevans.test() of spatstat is used to performs Clark-Evans test of aggregation for childcare centre in Choa Chu Kang planning area.

clarkevans.test(childcare_ck_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_ck_ppp
R = 0.90521, p-value = 0.1567
alternative hypothesis: two-sided
clarkevans.test(childcare_tm_ppp,
                correction="none",
                clipregion=NULL,
                alternative=c("two.sided"),
                nsim=999)

    Clark-Evans test
    No edge correction
    Z-test

data:  childcare_tm_ppp
R = 0.7994, p-value = 0.0002941
alternative hypothesis: two-sided

3.9 Second Order Spatial Point Pattern Analysis

3.9.1 Analysing Spatial Point Pattern using G- Function

The G function measures the distribution of the distances from an arbitrary event to its nearest event. In this section, I will compute G-function estimation by using Gest() of spatstat package. I will also perform monta carlo simulation test using envelope() of spatstat package.

3.9.1.1 Choa Chu Kang Planning Area
G_CK = Gest(childcare_ck_ppp, correction = "border")
plot(G_CK, xlim=c(0,500))

3.9.1.2 Performing Complete Spatial Randomness Test

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

We will use the Monte Carlo test with the G-function

G_CK.csr <- envelope(childcare_ck_ppp, Gest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_CK.csr)

3.9.1.3 Tampines Planning Area

We will replicate the code chunk above to perform the test for the Tampines Planning Area as well.

G_tm = Gest(childcare_tm_ppp, correction = "best")
plot(G_tm)

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

The code chunk below is used to perform the hypothesis testing.

G_tm.csr <- envelope(childcare_tm_ppp, Gest, correction = "all", nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(G_tm.csr)

3.9.2 Analysing Spatial Point Patterns Using F-function

The F function estimates the empty space function F(r) or its hazard rate h(r) from a point pattern in a window of arbitrary shape. In this section, we will compute F-function estimation by using Fest() of spatstat package, and the envelope()from before.

3.9.2.1 Choa Chu Kang

We will first compute the F-function estimation using Fest() of spatstat package

F_CK = Fest(childcare_ck_ppp)
plot(F_CK)

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

We will once again perform the Monte Carlo test with F-function:

F_CK.csr <- envelope(childcare_ck_ppp, Fest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_CK.csr)

3.9.2.2 Tampines Planning Area

We will do the same 2 steps for the Tampines Planning Area.

F_TM = Fest(childcare_tm_ppp)
plot(F_TM)

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected is p-value is smaller than alpha value of 0.001.

F_TM.csr <- envelope(childcare_tm_ppp, Fest, nsim = 999)
Generating 999 simulations of CSR  ...
1, 2, 3, ......10.........20.........30.........40.........50.........60..
.......70.........80.........90.........100.........110.........120.........130
.........140.........150.........160.........170.........180.........190........
.200.........210.........220.........230.........240.........250.........260......
...270.........280.........290.........300.........310.........320.........330....
.....340.........350.........360.........370.........380.........390.........400..
.......410.........420.........430.........440.........450.........460.........470
.........480.........490.........500.........510.........520.........530........
.540.........550.........560.........570.........580.........590.........600......
...610.........620.........630.........640.........650.........660.........670....
.....680.........690.........700.........710.........720.........730.........740..
.......750.........760.........770.........780.........790.........800.........810
.........820.........830.........840.........850.........860.........870........
.880.........890.........900.........910.........920.........930.........940......
...950.........960.........970.........980.........990........
999.

Done.
plot(F_TM.csr)

3.9.3 Analyising Spatial Point Patterns Using K-function

K-function measures the number of events found up to a given distance of any particular event. In this section, we will compute K-function estimates by using Kest() of spatstat package. We will once again use envelope() to perform the Monte Carlo test.

3.9.3.1 Choa Chu Kang Planning Area
K_ck = Kest(childcare_ck_ppp, correction = "Ripley")
plot(K_ck, . -r ~ r, ylab= "K(d)-r", xlab = "d(m)")

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

K_ck.csr <- envelope(childcare_ck_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_ck.csr, . - r ~ r, xlab="d", ylab="K(d)-r")

3.9.3.2 Tampines Planning Area
K_tm = Kest(childcare_tm_ppp, correction = "Ripley")
plot(K_tm, . -r ~ r, 
     ylab= "K(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

K_tm.csr <- envelope(childcare_tm_ppp, Kest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(K_tm.csr, . - r ~ r, 
     xlab="d", ylab="K(d)-r", xlim=c(0,500))

3.9.4 Analysing Spatial Point Patterns using the L function

Here, we will compute L-function estimation by using Lest() of spatstat package. We will also be performing the Monte Carlo Test.

3.9.4.1 Choa Chu Kang Planning Area
L_ck = Lest(childcare_ck_ppp, correction = "Ripley")
plot(L_ck, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)")

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Choa Chu Kang are randomly distributed.

H1= The distribution of childcare services at Choa Chu Kang are not randomly distributed.

The null hypothesis will be rejected if p-value if smaller than alpha value of 0.001.

L_ck.csr <- envelope(childcare_ck_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_ck.csr, . - r ~ r, xlab="d", ylab="L(d)-r")

3.9.4.2 Tampines Planning Area
L_tm = Lest(childcare_tm_ppp, correction = "Ripley")
plot(L_tm, . -r ~ r, 
     ylab= "L(d)-r", xlab = "d(m)", 
     xlim=c(0,1000))

To confirm the observed spatial patterns above, a hypothesis test will be conducted. The hypothesis and test are as follows:

Ho = The distribution of childcare services at Tampines are randomly distributed.

H1= The distribution of childcare services at Tampines are not randomly distributed.

The null hypothesis will be rejected if p-value is smaller than alpha value of 0.001.

L_tm.csr <- envelope(childcare_tm_ppp, Lest, nsim = 99, rank = 1, glocal=TRUE)
Generating 99 simulations of CSR  ...
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
99.

Done.
plot(L_tm.csr, . - r ~ r, 
     xlab="d", ylab="L(d)-r", xlim=c(0,500))